Save For Web originates with another product long ago that was incorporated into Photoshop. It was called ImageReady. ImageReady was all about making it easier to take files from Photoshop and make HTML friendly pages/images from it. Save For Web was a way of getting to ImageReady. Nowadays this functionality is all about templates. It's a quick and easy way of setting presets for gifs, jpegs, pngs, and other things as usually needed for web sites. That's why you see the option to strip out meta data, setting transparency for a gif file, convert to sRGB, etc. Those options are most useful when your exported file is going to be displayed on the web. You can do those things manually but Save For Web simplifies it a lot.
The regular Save As is just that. Saving a file format. It's up to you to set up the options you need for the saved file. It isn't any better or worse than Save For Web. If you're seeing better output from one than the other then I'd recommend experimenting with the quality option when saving.
As for the preferred option... there really isn't one. When I am exporting a picture for a model, for example, I'll usually do a regular save option because I don't know how she will use the picture. If I'm going to display on facebook or something then I'll usually use Save For Web with a preset just to simplify my process. Either way it's essentially the same result. Although with SFW I'll usually use much smaller files.
When it comes to printing, I always keep the highest quality file I can. That means the Photoshop file in 99% of cases.